Just as David Cameron and Huw Edwards would have wanted, Anthony and his workmates held a get-together to mark the royal wedding. They turned up at the gates of Buckingham Palace a full 24 hours beforehand. Suitably dressed up, they even had signs. Only these ones didn't have big pictures of Kate and Wills, but focused on things such as their own wages and crappy working conditions. And instead of getting their faces all over the BBC, Anthony's crew were moved on pretty sharpish by the police.

As you've already guessed, Anthony and his two-dozen-odd colleagues weren't just another bunch of tourists, descending on London for the big day. They turn up at Buckingham Palace almost every day to do the cleaning. And for months they have been fighting for a modest pay rise.

Instead of a just-above minimum wage of £6.45 an hour, they would like the London living wage of £7.85. The difference is small change to the royal household, which gets around £15m a year from taxpayers just for upkeep of the palaces. Nor, I imagine, would it make much odds to the firms contracted to do the cleaning. So what would a rise of £1.40 an hour mean to Anthony? "The difference between working two jobs a day – like I do at the moment – and having some time to see my children, or my friends, or going to college."

Here's something else you've worked out: Anthony's name is made up, in order to stop my interviewee from getting in trouble with his employer, and I can't be too specific about his living arrangements. But if you want to get an idea of what life on £6.45 an hour in London is like, don't start off with status or shopping. Instead, think about how the cleaners have to manage their time.

Anthony has to travel one-and-a-half hours from the palace back to his one-room bedsit, because he can't afford any accommodation closer in. Combine that with having to work two jobs, make his own lunch and rely on the vagaries of public transport, and he gets three hours' sleep a night. Straight after work on Fridays, he crawls into bed (often fully clothed), and doesn't wake up until late on Saturday afternoon.

While chatting with Anthony, I pick up his paperback. A fantasy novel, it has a dragon on the cover and more than 700 pages. He hasn't got very far, despite carrying it around for months. Why not? "I can't read more than 10 pages at a time, otherwise my eyes start to burn," he says. Which is what happens when you never get enough sleep, I suppose.

If the story of the princess and the pea had a moral, then so too does the tale of the princess and the underpaid cleaner. On the day of the wedding the air was thick with commentators over-reaching themselves to pronounce on What It All Said About Modern Britain. The crowds, that dress, the introduction of a commoner (as Kate Middleton and other products of £28,000-a-year Marlborough College almost certainly don't think of themselves as) into the royal family – all were camera fodder.

A few years ago, the London Health Observatory (LHO) produced a tube map with a difference – one that showed how long residents could expect to live. It began just by the Abbey, at Westminster, and ran out east. At each tube stop, local residents could expect to lose a year off their life expectancy. In the seven miles between Westminster and Canning Town, nearly seven years were shaved off the average male lifespan.

LHO recently updated the map to find that it now took two tube stops to knock off a year of life – but that may be a blip, or down to the amount of Olympic money poured into the East End. Certainly all the statistics show that the poorer you are in London, the worse your life expectancy.

Why? Think about Anthony's story. He can only afford cramped housing miles away from his workplace. He has the stress of juggling two jobs and of being treated like muck at the palace (male and female cleaners have to change in the same room, and hang their coats in the storage cupboard, so that they come out smelling of chemicals). His life chances, and those of his kids, too, are knocked back because of his wages.

In all likelihood, you've come across Anthony or someone like him – and barely spoken to them. They're the people who clean your office or your kids' school before you get in. They come in from corners of your city or town early in the morning or late at night and their lives are almost wholly cut off from yours.

"The new servile class," is how Danny Dorling, author of So You Think You Know About Britain, refers to them and he says they've grown out of all proportion in the past 25 years. Chris Hamnett, professor of geography at King's College, London, points out that economic segregation such as this didn't exist even in Victorian times.

"Did you watch any of the wedding?" I asked Anthony a few days afterwards. "What would I do that for?" That Friday, you see, was a day off, and Anthony uses his days off to catch up on sleep.